Properties

Label 10.9.c.b
Level 10
Weight 9
Character orbit 10.c
Analytic conductor 4.074
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.07378610061\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{601})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 8 + 8 \beta_{1} ) q^{2} + ( 22 - 21 \beta_{1} + \beta_{3} ) q^{3} + 128 \beta_{1} q^{4} + ( -216 - 71 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5} + ( 352 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} + ( 1442 + 1442 \beta_{1} - 21 \beta_{2} ) q^{7} + ( -1024 + 1024 \beta_{1} ) q^{8} + ( -1876 \beta_{1} + 43 \beta_{2} + 43 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 8 + 8 \beta_{1} ) q^{2} + ( 22 - 21 \beta_{1} + \beta_{3} ) q^{3} + 128 \beta_{1} q^{4} + ( -216 - 71 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5} + ( 352 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} + ( 1442 + 1442 \beta_{1} - 21 \beta_{2} ) q^{7} + ( -1024 + 1024 \beta_{1} ) q^{8} + ( -1876 \beta_{1} + 43 \beta_{2} + 43 \beta_{3} ) q^{9} + ( -1112 - 2272 \beta_{1} - 24 \beta_{2} + 72 \beta_{3} ) q^{10} + ( -3824 - 141 \beta_{1} + 141 \beta_{2} - 141 \beta_{3} ) q^{11} + ( 2816 + 2816 \beta_{1} - 128 \beta_{2} ) q^{12} + ( 11235 - 11553 \beta_{1} - 318 \beta_{3} ) q^{13} + ( 22904 \beta_{1} - 168 \beta_{2} - 168 \beta_{3} ) q^{14} + ( -28982 - 42167 \beta_{1} + 266 \beta_{2} - 153 \beta_{3} ) q^{15} -16384 q^{16} + ( 871 + 871 \beta_{1} + 66 \beta_{2} ) q^{17} + ( 15352 - 14664 \beta_{1} + 688 \beta_{3} ) q^{18} + ( 122890 \beta_{1} + 438 \beta_{2} + 438 \beta_{3} ) q^{19} + ( 9856 - 27264 \beta_{1} - 768 \beta_{2} + 384 \beta_{3} ) q^{20} + ( 221200 + 1883 \beta_{1} - 1883 \beta_{2} + 1883 \beta_{3} ) q^{21} + ( -30592 - 30592 \beta_{1} + 2256 \beta_{2} ) q^{22} + ( -113258 + 115139 \beta_{1} + 1881 \beta_{3} ) q^{23} + ( 44032 \beta_{1} - 1024 \beta_{2} - 1024 \beta_{3} ) q^{24} + ( -229705 - 172605 \beta_{1} - 435 \beta_{2} - 3045 \beta_{3} ) q^{25} + ( 179760 - 2544 \beta_{1} + 2544 \beta_{2} - 2544 \beta_{3} ) q^{26} + ( -220892 - 220892 \beta_{1} - 2836 \beta_{2} ) q^{27} + ( -184576 + 181888 \beta_{1} - 2688 \beta_{3} ) q^{28} + ( 132400 \beta_{1} + 96 \beta_{2} + 96 \beta_{3} ) q^{29} + ( 104256 - 567064 \beta_{1} + 3352 \beta_{2} + 904 \beta_{3} ) q^{30} + ( 1165300 - 2787 \beta_{1} + 2787 \beta_{2} - 2787 \beta_{3} ) q^{31} + ( -131072 - 131072 \beta_{1} ) q^{32} + ( -1143320 + 1133574 \beta_{1} - 9746 \beta_{3} ) q^{33} + ( 14464 \beta_{1} + 528 \beta_{2} + 528 \beta_{3} ) q^{34} + ( 746074 - 881356 \beta_{1} + 273 \beta_{2} + 14406 \beta_{3} ) q^{35} + ( 245632 + 5504 \beta_{1} - 5504 \beta_{2} + 5504 \beta_{3} ) q^{36} + ( -1386939 - 1386939 \beta_{1} - 3564 \beta_{2} ) q^{37} + ( -979616 + 986624 \beta_{1} + 7008 \beta_{3} ) q^{38} + ( 1899033 \beta_{1} + 4557 \beta_{2} + 4557 \beta_{3} ) q^{39} + ( 300032 - 145408 \beta_{1} - 9216 \beta_{2} - 3072 \beta_{3} ) q^{40} + ( 2677024 - 7653 \beta_{1} + 7653 \beta_{2} - 7653 \beta_{3} ) q^{41} + ( 1769600 + 1769600 \beta_{1} - 30128 \beta_{2} ) q^{42} + ( 1243038 - 1213605 \beta_{1} + 29433 \beta_{3} ) q^{43} + ( -471424 \beta_{1} + 18048 \beta_{2} + 18048 \beta_{3} ) q^{44} + ( -3054907 - 572642 \beta_{1} + 5021 \beta_{2} - 18098 \beta_{3} ) q^{45} + ( -1812128 + 15048 \beta_{1} - 15048 \beta_{2} + 15048 \beta_{3} ) q^{46} + ( -1398618 - 1398618 \beta_{1} + 72999 \beta_{2} ) q^{47} + ( -360448 + 344064 \beta_{1} - 16384 \beta_{3} ) q^{48} + ( 1646596 \beta_{1} - 60123 \beta_{2} - 60123 \beta_{3} ) q^{49} + ( -481160 - 3221960 \beta_{1} + 20880 \beta_{2} - 27840 \beta_{3} ) q^{50} + ( -457468 - 515 \beta_{1} + 515 \beta_{2} - 515 \beta_{3} ) q^{51} + ( 1438080 + 1438080 \beta_{1} + 40704 \beta_{2} ) q^{52} + ( 5047325 - 5097833 \beta_{1} - 50508 \beta_{3} ) q^{53} + ( -3556960 \beta_{1} - 22688 \beta_{2} - 22688 \beta_{3} ) q^{54} + ( -2351592 + 9825523 \beta_{1} - 51939 \beta_{2} - 1653 \beta_{3} ) q^{55} + ( -2953216 - 21504 \beta_{1} + 21504 \beta_{2} - 21504 \beta_{3} ) q^{56} + ( -596312 - 596312 \beta_{1} - 104056 \beta_{2} ) q^{57} + ( -1058432 + 1059968 \beta_{1} + 1536 \beta_{3} ) q^{58} + ( -172550 \beta_{1} + 127782 \beta_{2} + 127782 \beta_{3} ) q^{59} + ( 5377792 - 3675648 \beta_{1} + 19584 \beta_{2} + 34048 \beta_{3} ) q^{60} + ( -10824672 + 68331 \beta_{1} - 68331 \beta_{2} + 68331 \beta_{3} ) q^{61} + ( 9322400 + 9322400 \beta_{1} + 44592 \beta_{2} ) q^{62} + ( 9550534 - 9388029 \beta_{1} + 162505 \beta_{3} ) q^{63} -2097152 \beta_{1} q^{64} + ( 3874593 + 15997908 \beta_{1} + 79491 \beta_{2} + 103347 \beta_{3} ) q^{65} + ( -18293120 - 77968 \beta_{1} + 77968 \beta_{2} - 77968 \beta_{3} ) q^{66} + ( 2480678 + 2480678 \beta_{1} - 202113 \beta_{2} ) q^{67} + ( -111488 + 119936 \beta_{1} + 8448 \beta_{3} ) q^{68} + ( -9220477 \beta_{1} - 73757 \beta_{2} - 73757 \beta_{3} ) q^{69} + ( 13134688 - 1080072 \beta_{1} - 113064 \beta_{2} + 117432 \beta_{3} ) q^{70} + ( 4993028 - 108699 \beta_{1} + 108699 \beta_{2} - 108699 \beta_{3} ) q^{71} + ( 1965056 + 1965056 \beta_{1} - 88064 \beta_{2} ) q^{72} + ( -3126215 + 2677967 \beta_{1} - 448248 \beta_{3} ) q^{73} + ( -22219536 \beta_{1} - 28512 \beta_{2} - 28512 \beta_{3} ) q^{74} + ( -5516110 + 23912715 \beta_{1} + 96480 \beta_{2} - 284515 \beta_{3} ) q^{75} + ( -15673856 + 56064 \beta_{1} - 56064 \beta_{2} + 56064 \beta_{3} ) q^{76} + ( -27757240 - 27757240 \beta_{1} + 481026 \beta_{2} ) q^{77} + ( -15155808 + 15228720 \beta_{1} + 72912 \beta_{3} ) q^{78} + ( -34376880 \beta_{1} + 102768 \beta_{2} + 102768 \beta_{3} ) q^{79} + ( 3538944 + 1163264 \beta_{1} - 49152 \beta_{2} - 98304 \beta_{3} ) q^{80} + ( 24175343 + 120787 \beta_{1} - 120787 \beta_{2} + 120787 \beta_{3} ) q^{81} + ( 21416192 + 21416192 \beta_{1} + 122448 \beta_{2} ) q^{82} + ( 16164594 - 15967749 \beta_{1} + 196845 \beta_{3} ) q^{83} + ( 28072576 \beta_{1} - 241024 \beta_{2} - 241024 \beta_{3} ) q^{84} + ( -3095821 + 1235524 \beta_{1} - 17067 \beta_{2} + 3351 \beta_{3} ) q^{85} + ( 19888608 + 235464 \beta_{1} - 235464 \beta_{2} + 235464 \beta_{3} ) q^{86} + ( 2189536 + 2189536 \beta_{1} - 128272 \beta_{2} ) q^{87} + ( 3915776 - 3627008 \beta_{1} + 288768 \beta_{3} ) q^{88} + ( 70322120 \beta_{1} - 72360 \beta_{2} - 72360 \beta_{3} ) q^{89} + ( -20002904 - 28980224 \beta_{1} + 184952 \beta_{2} - 104616 \beta_{3} ) q^{90} + ( -17763396 - 215943 \beta_{1} + 215943 \beta_{2} - 215943 \beta_{3} ) q^{91} + ( -14497024 - 14497024 \beta_{1} - 240768 \beta_{2} ) q^{92} + ( 4700656 - 3652410 \beta_{1} + 1048246 \beta_{3} ) q^{93} + ( -21793896 \beta_{1} + 583992 \beta_{2} + 583992 \beta_{3} ) q^{94} + ( -20183500 - 36078750 \beta_{1} - 800850 \beta_{2} + 241650 \beta_{3} ) q^{95} + ( -5767168 - 131072 \beta_{1} + 131072 \beta_{2} - 131072 \beta_{3} ) q^{96} + ( 8470809 + 8470809 \beta_{1} + 115080 \beta_{2} ) q^{97} + ( -13653752 + 12691784 \beta_{1} - 961968 \beta_{3} ) q^{98} + ( 98005883 \beta_{1} - 422885 \beta_{2} - 422885 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 86q^{3} - 870q^{5} + 1376q^{6} + 5726q^{7} - 4096q^{8} + O(q^{10}) \) \( 4q + 32q^{2} + 86q^{3} - 870q^{5} + 1376q^{6} + 5726q^{7} - 4096q^{8} - 4640q^{10} - 14732q^{11} + 11008q^{12} + 45576q^{13} - 115090q^{15} - 65536q^{16} + 3616q^{17} + 60032q^{18} + 37120q^{20} + 877268q^{21} - 117856q^{22} - 456794q^{23} - 913600q^{25} + 729216q^{26} - 889240q^{27} - 732928q^{28} + 421920q^{30} + 4672348q^{31} - 524288q^{32} - 4553788q^{33} + 2956030q^{35} + 960512q^{36} - 5554884q^{37} - 3932480q^{38} + 1187840q^{40} + 10738708q^{41} + 7018144q^{42} + 4913286q^{43} - 12173390q^{45} - 7308704q^{46} - 5448474q^{47} - 1409024q^{48} - 1827200q^{50} - 1827812q^{51} + 5833728q^{52} + 20290316q^{53} - 9506940q^{55} - 11726848q^{56} - 2593360q^{57} - 4236800q^{58} + 21482240q^{60} - 43572012q^{61} + 37378784q^{62} + 37877126q^{63} + 15450660q^{65} - 72860608q^{66} + 9518486q^{67} - 462848q^{68} + 52077760q^{70} + 20406908q^{71} + 7684096q^{72} - 11608364q^{73} - 21302450q^{75} - 62919680q^{76} - 110066908q^{77} - 60769056q^{78} + 14254080q^{80} + 96218224q^{81} + 85909664q^{82} + 64264686q^{83} - 12424120q^{85} + 78612576q^{86} + 8501600q^{87} + 15085568q^{88} - 79432480q^{90} - 70189812q^{91} - 58469632q^{92} + 16706132q^{93} - 82819000q^{95} - 22544384q^{96} + 34113396q^{97} - 52691072q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 301 x^{2} + 22500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 151 \nu \)\()/150\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 250 \nu^{2} + 401 \nu + 37650 \)\()/50\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 375 \nu^{2} + 526 \nu - 56475 \)\()/75\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 5 \beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - \beta_{1} - 1506\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-151 \beta_{3} - 151 \beta_{2} + 2255 \beta_{1}\)\()/10\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(\beta_{1}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
11.7577i
12.7577i
11.7577i
12.7577i
8.00000 8.00000i −39.7883 39.7883i 128.000i −401.365 479.094i −636.612 144.447 144.447i −1024.00 1024.00i 3394.79i −7043.67 621.836i
3.2 8.00000 8.00000i 82.7883 + 82.7883i 128.000i −33.6352 + 624.094i 1324.61 2718.55 2718.55i −1024.00 1024.00i 7146.79i 4723.67 + 5261.84i
7.1 8.00000 + 8.00000i −39.7883 + 39.7883i 128.000i −401.365 + 479.094i −636.612 144.447 + 144.447i −1024.00 + 1024.00i 3394.79i −7043.67 + 621.836i
7.2 8.00000 + 8.00000i 82.7883 82.7883i 128.000i −33.6352 624.094i 1324.61 2718.55 + 2718.55i −1024.00 + 1024.00i 7146.79i 4723.67 5261.84i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.9.c.b 4
3.b odd 2 1 90.9.g.a 4
4.b odd 2 1 80.9.p.a 4
5.b even 2 1 50.9.c.b 4
5.c odd 4 1 inner 10.9.c.b 4
5.c odd 4 1 50.9.c.b 4
15.e even 4 1 90.9.g.a 4
20.e even 4 1 80.9.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.9.c.b 4 1.a even 1 1 trivial
10.9.c.b 4 5.c odd 4 1 inner
50.9.c.b 4 5.b even 2 1
50.9.c.b 4 5.c odd 4 1
80.9.p.a 4 4.b odd 2 1
80.9.p.a 4 20.e even 4 1
90.9.g.a 4 3.b odd 2 1
90.9.g.a 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 86 T_{3}^{3} + 3698 T_{3}^{2} + 566568 T_{3} + 43401744 \) acting on \(S_{9}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 16 T + 128 T^{2} )^{2} \)
$3$ \( 1 - 86 T + 3698 T^{2} + 2322 T^{3} - 43400286 T^{4} + 15234642 T^{5} + 159186774258 T^{6} - 24288940137366 T^{7} + 1853020188851841 T^{8} \)
$5$ \( 1 + 870 T + 835250 T^{2} + 339843750 T^{3} + 152587890625 T^{4} \)
$7$ \( 1 - 5726 T + 16393538 T^{2} - 37506290598 T^{3} + 85192723481474 T^{4} - 216216301545640998 T^{5} + \)\(54\!\cdots\!38\)\( T^{6} - \)\(10\!\cdots\!26\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 7366 T + 143570226 T^{2} + 1578967517446 T^{3} + 45949729863572161 T^{4} )^{2} \)
$13$ \( 1 - 45576 T + 1038585888 T^{2} - 14387574925368 T^{3} - 50735886714949186 T^{4} - \)\(11\!\cdots\!28\)\( T^{5} + \)\(69\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!36\)\( T^{7} + \)\(44\!\cdots\!81\)\( T^{8} \)
$17$ \( 1 - 3616 T + 6537728 T^{2} - 25111917401568 T^{3} + 96455844642935681534 T^{4} - \)\(17\!\cdots\!88\)\( T^{5} + \)\(31\!\cdots\!68\)\( T^{6} - \)\(12\!\cdots\!36\)\( T^{7} + \)\(23\!\cdots\!61\)\( T^{8} \)
$19$ \( 1 - 31965435764 T^{2} + \)\(65\!\cdots\!86\)\( T^{4} - \)\(92\!\cdots\!84\)\( T^{6} + \)\(83\!\cdots\!61\)\( T^{8} \)
$23$ \( 1 + 456794 T + 104330379218 T^{2} + 35544578018872962 T^{3} + \)\(12\!\cdots\!94\)\( T^{4} + \)\(27\!\cdots\!22\)\( T^{5} + \)\(63\!\cdots\!98\)\( T^{6} + \)\(21\!\cdots\!54\)\( T^{7} + \)\(37\!\cdots\!21\)\( T^{8} \)
$29$ \( 1 - 1965649191044 T^{2} + \)\(14\!\cdots\!26\)\( T^{4} - \)\(49\!\cdots\!24\)\( T^{6} + \)\(62\!\cdots\!41\)\( T^{8} \)
$31$ \( ( 1 - 2336174 T + 2953504595226 T^{2} - 1992501866502690734 T^{3} + \)\(72\!\cdots\!81\)\( T^{4} )^{2} \)
$37$ \( 1 + 5554884 T + 15428368126728 T^{2} + 40407042725082894252 T^{3} + \)\(91\!\cdots\!94\)\( T^{4} + \)\(14\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!48\)\( T^{6} + \)\(24\!\cdots\!24\)\( T^{7} + \)\(15\!\cdots\!81\)\( T^{8} \)
$41$ \( ( 1 - 5369354 T + 22297350707346 T^{2} - 42873890218681757834 T^{3} + \)\(63\!\cdots\!41\)\( T^{4} )^{2} \)
$43$ \( 1 - 4913286 T + 12070189658898 T^{2} - 40277437086392745918 T^{3} + \)\(12\!\cdots\!94\)\( T^{4} - \)\(47\!\cdots\!18\)\( T^{5} + \)\(16\!\cdots\!98\)\( T^{6} - \)\(78\!\cdots\!86\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 5448474 T + 14842934464338 T^{2} - 68165833176447628158 T^{3} - \)\(10\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!38\)\( T^{5} + \)\(84\!\cdots\!98\)\( T^{6} + \)\(73\!\cdots\!94\)\( T^{7} + \)\(32\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 - 20290316 T + 205848461689928 T^{2} - \)\(19\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!74\)\( T^{4} - \)\(11\!\cdots\!68\)\( T^{5} + \)\(79\!\cdots\!88\)\( T^{6} - \)\(48\!\cdots\!96\)\( T^{7} + \)\(15\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 - 96598605716084 T^{2} + \)\(45\!\cdots\!46\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{6} + \)\(46\!\cdots\!81\)\( T^{8} \)
$61$ \( ( 1 + 21786006 T + 431918528798546 T^{2} + \)\(41\!\cdots\!86\)\( T^{3} + \)\(36\!\cdots\!61\)\( T^{4} )^{2} \)
$67$ \( 1 - 9518486 T + 45300787866098 T^{2} - \)\(10\!\cdots\!58\)\( T^{3} - \)\(62\!\cdots\!26\)\( T^{4} - \)\(42\!\cdots\!78\)\( T^{5} + \)\(74\!\cdots\!38\)\( T^{6} - \)\(63\!\cdots\!06\)\( T^{7} + \)\(27\!\cdots\!61\)\( T^{8} \)
$71$ \( ( 1 - 10203454 T + 1140007205044026 T^{2} - \)\(65\!\cdots\!94\)\( T^{3} + \)\(41\!\cdots\!21\)\( T^{4} )^{2} \)
$73$ \( 1 + 11608364 T + 67377057378248 T^{2} - \)\(79\!\cdots\!48\)\( T^{3} - \)\(12\!\cdots\!46\)\( T^{4} - \)\(64\!\cdots\!88\)\( T^{5} + \)\(43\!\cdots\!28\)\( T^{6} + \)\(60\!\cdots\!24\)\( T^{7} + \)\(42\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 - 3387529564746244 T^{2} + \)\(67\!\cdots\!26\)\( T^{4} - \)\(77\!\cdots\!24\)\( T^{6} + \)\(52\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 - 64264686 T + 2064974933339298 T^{2} - \)\(15\!\cdots\!58\)\( T^{3} + \)\(12\!\cdots\!74\)\( T^{4} - \)\(35\!\cdots\!78\)\( T^{5} + \)\(10\!\cdots\!38\)\( T^{6} - \)\(73\!\cdots\!06\)\( T^{7} + \)\(25\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 - 5698613213739524 T^{2} + \)\(37\!\cdots\!66\)\( T^{4} - \)\(88\!\cdots\!64\)\( T^{6} + \)\(24\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 - 34113396 T + 581861893326408 T^{2} - \)\(26\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - \)\(21\!\cdots\!28\)\( T^{5} + \)\(35\!\cdots\!68\)\( T^{6} - \)\(16\!\cdots\!76\)\( T^{7} + \)\(37\!\cdots\!41\)\( T^{8} \)
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